Calculus homework help

1. Let f(x) = 3x. Tabulate the change of f over the intervals (i) [1, 3],

(ii)[1, 2], (iii)[1, −1], (iv) [1, 1.2], (v)[1, 1.1], (vi)[1, 1.01]. Graph y = 3x

together with the secant line passing through (1, f(1)) and (1.1, f(1.1)).

interval [1, 3] [1, 2] [1, −1] [1, 1.2] [1, 1.1]

average rate of change

Based on the pattern of numbers (without deriving a general formula),

estimate how quickly f(x) = 3x changes at x = 1.

2. Let f(x) = ex where e = 2.71. Tabulate the change of f over the

intervals(i) [0, 2], (ii)[0, 1], (iii)[0, .5], (iv) [0, 0.25], (v)[0, 0.1].

interval [0, 2] [0, 1] [0, 0.5] [0, 0.25] [0, 0.1]

average rate of change

Estimate the rate of change of f(x) = ex at x = 0. Check your answer

with this video:

3. Cyclists A and B rode along a straight one way road starting from

the same location and they rode for one hour. Cyclist A rode half

the time with speed 10 miles/hour and the other half time with speed

30 miles/hour. Cyclist B rode half the total distance with speed 10

miles/hour and half the distance with speed 30/hours.

(a) Calculate the total distance traveled by cyclist A. Graph the ve-

locity function for cyclist A and calculate the average trip velocity for

cyclist A.

(b) Calculate the total distance traveled by cyclist B. Calculate the av-

erage velocity for cyclist B. Graph the velocity function for cyclist B.

4. Use the limit definition of the rate of change to calculate how quickly

f(x) = x2 changes at x = −4.

5. Go to a financial website (for exmaple, finance.google.com), pick your

favorite stock. By p(t) denote the price at which the stock was exchanged

at time t where t is measured in seconds from last Friday midday. What

does p(0) mean? What does p(7200) mean? Estimate the average rate

of change of the stock for the following time intervals

time interval [0, 7200] [0, 3600] [0, 360] [0, 180] [0, 60]

average rate of change

Based on your calculations do you think the instantaneous rate of change

of the price p(t) exist? Can you briefly describe in words the behavior

of p(t) near t = 0.

6. youtube.com/watch?v=zrpjns9rk3Y Join Professor Goetz on his glider

flight from Pedra Bonita, Rio, Brazil to the beach. While watching it try

to match these graphs obtained by the GPS. Then answer the following

questions:

(a) What was the total descent of the flight? How long did it last? 1

2

(b) What was the average rate of descent throughout the flight? (What

are the units?)

(c) What was the maximum instantaneous rate of descent during the

flight? At what time did it happen?

(d) Was there a time interval when the glider was ascending?

7. temporary removed –

8. If a house bought for $1000000 appreciates at an annual percentage rate

of 20%, what is the average dollar rate of change of its price during (a)

the first year, (b) the second year, (c) the third year? What are the

units of these rates of change?

9. Discuss on the forum but no need to submit

(a) The Standard Poor’s/Case-Shiller 20-city housing index fell 16.3%

in July 2008 from a year earlier.

(b) Home prices were tumbling by the sharpest annual rate ever in July

2008.

(c) However, the rate of monthly declines was slowing in July 2008.

Denote P(t) – the value of the index at time t, Rannual(t) – annual rate

of P(t) and Rmonthly(t) – monthly rate of P(t). Write down the formulas

for Rannual(t) and Rmonthly(t) terms of P(t). Explain and interpret (a),

(b), (c).

1. Quiz There is no need to

submit this quiz.

However, please mark

your choices, and then

check your answers on the

homework forum after the

date due.

1. Let f(x) = x3. The average rate of change of f on the interval [−2, 2] is

(A) 16

(B) 3×2

(C) 0

(D) 4

2. The tangent line to y = f(x) at (3, 5) has slope

(A) 3

(B) 5

(C) f(5)−f(3)

5−3

(D) limb→3 f(b)−f(3)

b−3 .

3. The tangent line to y = x + 2 at (3, 5) has slope

(A) 1

(B) 2

(C) 3

(D) 5 .

4. The tangent line to y = −(x − 1)2 at (1, 0) has slope

(A) −1

(B) 0

(C) 1

(D) 2 .

3

5. Based on the graph of f below,

b

1

1

what is the instantaneous rate of change of f at x = 1?

(A) 0

(B) 1

(C) 2

(D) Between 1 and 2.

6. Based on the graph below,

b

a

the instantaneous rate of change of f at x = a

(A) is positive

(B) is negative

(C) f(a)

(D) does not exist.

7. This chart from Altos Research LLC embedded image: http://charts.altosresearch.com/altos/app?s=medianra=cq=ast=CAcid=306z=asz=sts

illustrates the price of a single family home P(t) as a function of time.

We can conclude the following about the changes of P(t) over the period

of one year marked on the horizontal axis.

(A) the change of P(t) is -$830000 and the relative change is about -10%

(B) the change of P(t) is $25000 and the relative change is about 3%

(C) the change of P(t) is $80000 and the relative change is about $25000

(D) none of the above.

8. An object is moving along a straight horizontal line as follows. It starts

at x = 0 and then in the first second it moves to the right to x = 2.

Then in the second second the objects moves to the left to x = −1, and

4

finally in the third second to the right to stop at x = 1. Which of the

graphs is a possible graph of the velocity function?

(A)

1

1

y = f(t)

v

t

(B)

1

1

v = f(t)

v

t

(C)

5

1

1

v = f(t)

v

t

(D)

1

1

v = f(t)

v